The mural — which depicts a young girl searching a soldier — is located on grounds belonging to St. Michael’s Hotel, opposite the Jacir Palace Hotel.
Police cordoned off the site and reported that drilling had taken place around the mural.
They said they have opened an investigation into the incident.
Two suspects were taken in for questioning, but they have since been released.
Police spokesperson Luay Izriqat told Ma’an that officials from the Tourism and Antiquities police department had investigated the site and have taken necessary steps to preserve the artwork.
He called on Palestinians to help protect their cultural heritage.
Deputy-head of the Bethlehem municipality, Issam Ijha, confirmed that the drilling work indicates that the perpetrators were intending to remove and sell the artwork.
He said that the municipality had sent a formal letter to the Ministry of Tourism to demand protection for Banky’s murals in Bethlehem under a Jordanian law relating to antiquities that applies in Palestine.
The process will take some time before it is approved, Ijha added.
The governor of Bethlehem has issued an order preventing the removal of the mural from its location next to St. Michael’s Hotel.
The British graffiti artist painted at least 12 murals across Bethlehem and Beit Sahour on the Israeli separation wall and on the sides of Palestinian buildings and shops, and his images have become part of the local landscape.
Tourists are regularly taken to see the artworks.
One satirical mural, depicting an Israeli soldier checking a donkey’s identity papers, was painted over by Bethlehem residents in 2007 after locals interpreted the image as offensive.
In 2012, five works by Banksy, including several from Bethlehem, were exhibited in Miami after having been removed by the owners of the buildings on which they were painted.
One of the men involved in shipping the work to the United States tried to sell the works for up to $750,000, but was prevented from doing so after Banksy issued a statement criticizing the gallery owner.
Banksy is believed to have started out as a graffiti artist in London, although his identity remains shrouded in mystery.
In recent months, he painted a number of works in Gaza as a damning critique of Israel’s bombardment of the coastal enclave, which destroyed or damaged more than 100,000 homes and killed nearly 2,200 Palestinians, most of them civilians.
Mirna al-Atrash contributed reporting.